QMGreensite_merged

6.2. THEHEISENBERGUNCERTAINTYPRINCIPLE 91

Thetotalenergyisminimizedford/dR=0,andthisminimumisobtained
at

R=

3 ̄h^2

4 me^2

(6.32)

whichisnotveryfarofftheBohrradius

r 1 =

̄h^2

me^2

(6.33)

Wecannowunderstandwhytheelectrondoesn’tfallintothenucleus. Thepo-
tentialenergyisminimized byanelectron localizedat r =0. However,themore
localizedanelectronwavefunctionis,thesmaller∆xis,andthesmaller∆xis,the
greateristheuncertaintyin∆p. Butthegreaterthevalueof∆p,thegreateristhe
expectationvalueofkineticenergy

<

p^2

2 m

>∼

̄h^2

2 m∆x^2

(6.34)

andatsomepointthis(positive)kineticenergyoverwhelmsthe(negative)Coulomb
potential,whichgoeslike−e^2 /∆x. Thisiswhytheminimumelectronenergyinthe
Hydrogenatomisobtainedbyawavefunctionofsomefiniteextent,ontheorderof
theBohrradius.
Theapplicationofthe UncertaintyprincipletotheHydrogen atomshowsthat
thereismuchmoretothisprinciplethansimplythefactthatanobservationdisturbs
anobservedobject. Thereareverymanyhydrogenatomsintheuniverse;veryfew
of themare underobservation. If∆pwere dueto a disturbanceby observation,
thentherewouldbenothingtopreventanunobservedelectronfromfallingintothe
nucleus. Nevertheless, allhydrogenatoms, observedor not,have astableground
state,andthisisduetothefactthatthereisnophysicalstateinwhichanelectron
isbothlocalized,andatrest.

Problem:Computetheexpectationvalueofthekineticandpotentialenergyfrom
eq. (6.26),usingthegaussianwavepacketinthreedimensions

φ(x,y,z)=Nexp[−(x^2 +y^2 +z^2 )/ 2 R^2 ] (6.35)

(applythenormalizationconditiontodetermineN). Incomputingtheexpectation
valueofthepotentialenergy,theformulaforsphericalcoordinates
∫
dxdydzf(r)= 4 π

∫

drr^2 f(r) (6.36)

maybeuseful.FindthevalueofRwhichminimizes,andcomparethisvalue
totheBohrradius. Alsocomparenumericallytheminimumvalueoftothe
groundstateenergyE 1 oftheBohratom.